Sistem Komunikasi II · 3.1. Introduction Encoder Modulator RF Modulator Decoder Demodulator & Detector RF Demodulator 100101… 100101… 10101… 1011… Filtering < Mapping < Detection

Sistem Komunikasi II(Digital Communication Systems)

Lecture #3: Demodulasi / Deteksi Baseband(Baseband Demodulation / Detection)

- PART I –

Topik:3.1 Pendahuluan.

3.2 Representasi Geometris dari Sinyal.

3.3 Optimal Detection: “Maximum Likelihood Detection”.

3.4 Energy/Symbol, Energy/Bit, and Minimum Distance.

3.5 Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Lkelihood Detection.

3.6 Optimal Filter: “Matched Filter” or “Correlator

3.1. Introduction

Encoder ModulatorRF

Modulator

DecoderDemodulator& Detector

RF Demodulator

100101…

100101…

10101…

1011…

Filtering <Mapping <

Detection <

Block Diagram dari Sistem Komunikasi Digital:Transmitter

Receiver

noise

Kanal

3.1. Pendahuluan – cont.

mi adalah simbol digit yang me-representasikan informasi digital (message).

1 2[ , ,..., ]i M Alfabet simbolm m m m∈ ←

Transmiter Receiver(Channel)mi Si (t) x(t)

Simbol Digit

n(t)

Estimasi of mi

Contoh:1. Binary PAM: m1 = 0, m2 = 12. 4-ary PAM: m1 = 00, m2 = 01 , m3 = 10 , m4 = 11

ˆ im10101… 10101…

Sistem Komunikasi Digital (Baseband):

Simbol Waveform

3.1. Pendahuluan – cont.

Transmiter Receiver

Kanal AWGNmi Si (t) x(t)

Simbol Digit

n(t)

Estimasi of mi

ˆ im10101… 10101…

Sistem Komunikasi Digital (Baseband):

Simbol Waveform

Filter (Demod) Decision

sampling st kT=x(t) ˆ im

DetectionFiltering

⊕

( ) is White Gaussian Noise (WGN).n t

Goals:1. Menentukan bentuk filtering yang

optimal.

2. Menentukan bentuk detection yang optimal.

3.2. Representasi Geometris dari Sinyal

Representasi Geometris dari sinyal si(t) :

1

0( ) ( )1,2,...,

Ni ij j

j

t Ts t s ti M

φ=

≤ ≤= ⋅=∑

0

1,2,...,( ) ( )1,2,...,

Tij i j

i Ms s t t dtj N

φ

== ⋅=∫

( ) ; 1, 2,...,j t j Nφ =

Ekspansi

Koefisien Ekspansi

Fungsi Basis Orthonormal

0

1 ;( ) (: )

0 ;

T

i ji j

t tOrthonormal dti j

φ φ=

⋅ = ≠∫

Sintesis

Analisis

N M≤

3.2. Representasi Geometris dari Sinyal – cont.

( )is t

0

T

d t∫

0

T

d t∫

0

T

d t∫

1 ( )tφ

2 ( )tφ

( )N tφ

1is

2is

i Ns

Analisis: Sintesis:

∑

1 ( )tφ

2 ( )tφ

( )N tφ

1is

2is

i Ns

( )is t

01, 2,...,( ) ( ) ;ij i j

Tj Ns s t t dtφ == ⋅∫ 1

0( ) ( ) ;i ij j

N

jt Ts t s tφ

=≤ ≤= ⋅∑


Contoh: Binary PAM (NRZ)

0

1 ;( ) ( )

0 ;

T

i ji j

t t dti j

φ φ=

⋅ = ≠∫

Secara intuitif … fungsi basis:

T

A

s1(t)

-A

s2(t)

T

Tapi, K = ?

T

K

( )tφ

2

0

( ) 1T

i t dtφ =∫

2 2

0

( )T

t dt K Tφ =∫1KT

=

Ingat …

m1 = 1 m2 = 0


Contoh: Binary PAM – cont.

1 1

0

( ) ( )T

t t dt A Ts s φ= ⋅ =∫

T

A

S1(t)

-A

S2(t)

T

T

( )tφ1T

Analisis (koefisien ekspansi):

2 2

0

( ) ( )T

t t dt A Ts s φ= ⋅ = −∫

Sintesis:

1 1( ) ( ) ( )t t A T ts s φ φ= ⋅ = ⋅

2 2( ) ( ) ( )t t A T ts s φ φ= ⋅ = − ⋅

Fungsi Basis:

m1 = 1 m2 = 0


Contoh: Binary PAM – cont.

1 A Ts =

T

A

s1(t)

-A

s2(t)

T

T

( )tφ1T

Representasi Geometris:

2 A Ts = −

( )tφ

Fungsi Basis:

1s2s

A TA T− 0

Signal Space (Konstelasi Sinyal)1-Dimension (1D)

1 fungsi basis

si(t) si ~ sample

m1 = 1 m2 = 0


2

0

( ) 1T

i t dtφ =∫

1KT

=

Fungsi Basis( )tφ

K

T

m1 = 00, m2 = 01 m3 = 10, m4 = 11

Contoh: M-ary PAM (M=4)

s4(t)

-A

Tt

T

s2(t)

A/3

t

s3(t)

T-A/3

tT

s1(t)

A

t

t


4

3

2

1

1

0

2

0

3

0

4

0

( ) ( )

( ) ( )3

( ) ( )3

( ) ( )

T

T

T

T

t t dt A T

A Tt t dt

A Tt t dt

t t

s

dt A T

s

s

s

s

ss

s

φ

φ

φ

φ

= ⋅ = −

= ⋅ = −

= ⋅ =

= ⋅ =

∫

∫

∫

∫

Analisis (koefisien ekspansi):

Contoh: M-ary PAM (M=4) – cont.

Signal Space (Konstelasi Sinyal)1-Dimensi (1D)

1 Fungsi Basis

si(t) si ~ sample

Representasi Geometris:

0

3s4s 2s 1s

A T−3A T

−3A T A T

( )tφ


Point-point penting:

Sinyal Waveform‘dipetakan’ menjadi Sinyal Vektor

is( )i ts ; 1, 2,...,i M=

1, 2, ...;( )j j Ntφ =Fungsi basis berperan sebagai fungsi pemetaaan tersebut.

Fungsi basis bersifat Orthonormal:

2

0 0

1 ;( ) ( ) ( ) 1

0 ;

T T

i j i

i jt t dt t dt

i jφ φ φ

=⋅ = = ⇒

≠∫ ∫

3.3. Optimal Detection: “Maximum Likelihood Detection”

si (t) x(t)

n(t)

+

+⊕

Channel

DetectionFiltering

ˆ imFilter (Demod) Decision

sampling st kT=

z(t) z(kTs )

Baseband (PAM )Demodulation & Detection

x(t) = si (t) + n(t) 0

sT

d t∫

1

sT

( )tφ

Ts

ˆ imDecision

sampling st kT=

z(t) z(kTs )

DetectionMapping

3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.

x(t) = s(t) + n(t) 0

sT

d t∫m̂

Decision

sampling st kT=

z(t)

sample(test statistics)

Detection

MAPPING : Waveform Sample

z(kTs )

Mapping

1

sT

( )tφ

Ts

Konstelasi Sinyal (untuk Binary PAM NRZ):

Binary PAM NRZ

Ts

A

s1(t)

1 1m =-A

s2(t)

Ts

2 0m =

( )tφ1s2s

sA TsA T− 0


x(t) = s(t) + n(t) Decision

DetectionMapping

Konstelasi Sinyal (untuk Binary PAM NRZ):

( )tφ1s2s

sA TsA T− 0

Binary PAM NRZ

DECISION : Bandingkan test statistic VS. sebuah nilai threshold.

Ts

A

s1(t)

1 1m =

sampling st kT=

z(t) z(kTs ) ( )sz kTm̂

<

<1H

2Hλ

-A

s2(t)

Ts

2 0m =

0

sT

d t∫

1

sT

( )tφ

Ts


( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

0

0 0

0

s

s

s s

t = kT

t = kT

t = kT t = kT

s

s s

s

T

T T

i

i

z kT z t

x t t dt

s t t dt n t t dt

s k n k

φ

φ φ

=

=

= +

= +

∫

∫ ∫

2igaussian random variable ~ N(s , )nσ

2gaussian random variable ~ N(0, )nσ

mean variance0iz s n= +

( ) ( ) ( )ix t s t n t= +


0

0

0

0 2

2121( )

2e

n

nn

p nσ

πσ

=

0 0n

1 sTs A=2 sTs A= − 0 z

0

2

02

22

12

2

1( | ) en

nz s

p z sσ

πσ

−

=0

2

01

12

12

2

1( | ) en

nz s

p z sσ

πσ

−

=

P D F o f W G N

2Conditional PDF of ( dikirim) z s 1Conditional PDF of ( dikirim)z s


1 sTs A=2 sTs A= −

0 z

2( | )p z s 1( | )p z s

Likelihood R atio T est:

0z

0 1( | )p z s

0 2( | )p z s

Likelihood s1Likelihood s2

( )( )

0 10

0 2

|( )

|p z s

zp z s

Λ = <

<1H

2H

( )( )

1

2

p sp s

1 1

2 2

H s d ik irim .

H s d ik irim .

=

=1



( )( )

10

2

|( )

|p z s

zp z s

Λ = <

<1H

2H

( )( )

1

2

1p sp s

=

0

0

0

0

2

2

2

1

2

2

1

2

1

2

2

2

1 exp

1 exp

n

n

n

n

z s

z s

πσ

πσ

σ

σ

− =

−

<<1H

2H

1

0 0 0

0 0 0

2 20 0 11

2 2 2

2 20 0 22

2 2 2

2exp exp exp

2 2 2

2exp exp exp

2 2 2

n n n

n n n

z z ss

z z ss

σ σ σ

σ σ σ

− ⋅ − ⋅ −

= − ⋅ − ⋅ −

<

<1H

2H1

( ) ( )1 2

Untuk 'equi-probable' binary simbol digit:

1/ 2p s p s= =



<

<1H

2H

1

<<1H

2H[ ]ln 1 0=

1 2

2s s+

0z <

<1H

2H

Maximum Likelihood (ML) Detection Ruleuntuk Transmisi Binary PAM

( )0 0

2 20 1 2 1 2

0 2 2

( ) ( )exp2n n

z s s s szσ σ

− −Λ = −

( )0 0

2 20 1 2 1 2

0 2 2

( ) ( )ln2n n

z s s s szσ σ− −

Λ = −

0

0 1 22

( )

n

z s sσ−

<

<1H

2H 0

2 21 2

2

( )2 n

s sσ−



Contoh: Binary PAM

1 2 ( ) 0

2 2s sA T A Ts s + −+

= =0z <

<1H

2H

Konstelasi Sinyal:

Ts

A

s1(t)

1 1m =-A

s2(t)

Ts

2 0m =

( )tφ1s2s

sA TsA T− 0


x(t) = s(t) + n(t) 0

sT

d t∫

1

sT

( )tφ

Ts

Decision

MLDetectionMappingBinary PAM NRZ

Ts

A

s1(t)

1 1m =-A

s2(t)

Ts

2 0m =

sampling st kT=

z(t) z(kTs ) zm̂

<

<1H

2H0

( )tφ1s2s

sA TsA T− 0

Decision Region IDecision Region II

(Optimal) Maximum Likelihood Detection untuk Binary PAM NRZ:



Contoh: 4-Ary PAM

4 3 ( 3) 2 2 2 3

s ss

A T A Ts sA T

− + −+= = −0z <

<3H

4H

0

3s4s 2s 1s

sA T−

3sA T

−3

sA T sA T( )tφ

3 2 3 3 0

2 2s sA T A Ts s − ++

= =0z <

<2H

3H

2 1 3 2 2 2 3

s ss

A T A Ts s A T++

= =0z <

<1H

2H

3 Nilai Threshold


x(t) = s(t) + n(t) 0

sT

d t∫

1

sT

( )tφ

Ts

Decision

MLDetection

Mapping

sampling st kT=

z(t) z(kTs )

z

m̂

<

<3H

4H

23 sA T−

(Optimal) Maximum Likelihood Detection untuk M-ary PAM:

z <

<2H

3H0

z <

<1H

2H

23 sA T

0

3s4s 2s 1s

sA T−3

sA T−

3sA T

sA T( )tφ

IIIII IIV


x(t) = s(t) + n(t) 0

sT

d t∫

1

sT

( )tφ

Ts

MLDetectionFungsi : Mapping

Hardware: Correlator

zDecisionsampling st kT=

z(t) z(kTs )m̂

<<1H

2H0

Correlator Receiver dengan ML Detection untuk Binary PAM NRZ

z

3.4. Energy/Symbol, Energy/Bit, dan Minimum Distance.

1

min

.;M = Jumlah simbol di dalam alfabetEnergy/Symbol,

Energy/Bit,

Minimum Distance,

1

jarak antara 2 simbol yang terdekat .

k

M

s sk

bb s

s

E EM

TE E

T

D

=

=

=

=

∑

2min , 2

2

b s s s

b

E E A T D A T

E

= = =

=

( )tφ1s2s

sA TsA T− 0

Contoh: Binary PAM NRZ

Ts

A

s1(t)

1 1m =-A

s2(t)

Ts

2 0m =

Konstelasi Sinyal:

3.4. Energy/Symbol, Energy/Bit, dan Minimum Distance – cont.

Contoh: 4-ary PAM

2 2min

5 1 5 2 , = , 9 2 18 3

2 = 25

s s b s s s

b

E A T E E A T D A T

E

= = =

s4(t)

-A

Tst

Ts

s2(t)

A/3t

s3(t)

Ts-A/3t

Ts

s1(t)

A

t

0

3s4s 2s 1s

sA T−3

sA T−

3sA T sA T

( )tφKonstelasi Sinyal:

3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection

1 2

1 2

2

1 1 2 2

2

( | ) ( | ) ( | ) ( | )

s sz

s sz

P e s p z s dz P e s p z s dz

+=

∞

+−∞ =

= =∫ ∫

2( | )p z s 1( | )p z s

Probabilitas Error:

1 2

2s s+

z

1 1 2 2( | ) ( ) ( | ) ( )eP P e s P s P e s P s= ⋅ + ⋅ - Probabilitas Total Rata2

[ ]1 21 ( | ) ( | )2P e s P e s= + - equi-probable simbol digit

2 1( | ) ( | )P e s P e s= = - conditional PDF simetrik

1 2

2s s+z <

<1H

2H

ML Detection

2s 1s

3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection – cont.

1 2 1 2 0 0

2

22

2 2

1 1( | ) exp22e

s s s s n nz z

z sP p z s dz dzσσ π

∞ ∞

+ += =

− = =

∫ ∫

0

0 0

2n

1 n n

z s duu du dzdz

σσ σ

−= ⇒ = → =

- pergantian variabel

1 2

0

2

2

1 1 exp22

n

s su

u du

σ

π

∞

−=

= ∫

- Complementary Error Function (Q-Function) - ditabulasikan -

1 2 0

0

2 1 2e

2

1 1P exp Q2 22

n

s s n

s su du

σ

σπ

∞

−

− = = ∫

Probabilitas Error:

3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection – cont.

Probabilitas Error:

Contoh: Binary PAM

0

0

0

1 2e

min

0 0 0

P Q2

( ) Q

2

2 2 Q Q Q Q

2

n

s s

n

s s b

n

s s

A T A T

A T E E DN N N

σ

σ

σ

−=

− −=

= = = =

( )tφ1s2s

sA TsA T− 0

Symbol-Error Rate (SER)Bit-Error Rate (BER)

3.6. Optimal Filter: “Matched Filter” or “Correlator”

( )Respon Impuls: ( )sh t s T t= −

,dimana ( ) adalah sinyal input, adalah durasi dari s(t).ss t T

Kriteria optimal untuk filtering:

Bentuk demodulator filter yang optimal adalah filter yang me-maksimalkan Signal-to-Noise Power Ratio (SNR) pada output-nya.

Filter yang memenuhi kriteria di atas: Matched Filter

Untuk Binary PAM NRZ:

Ts

Ah(t)

ˆ imMatched Filter Decision

sampling st kT=

z(t) z(kTs )⊕Si (t)

n(t)

DetectionFiltering

3.6. Optimal Filter: “Matched Filter” or “Correlator” – cont.

Matched Filter sebagai Correlator – cont.

( )x t ix0

sT

d t∫

( )tφ

ix( )x t

( ) ( )sh t T tφ= −

h(t)

st kT=

Matched FilterCorrelator

Ekuivalensi antara Correlator dan Matched Filter :

0

( ) ( )sT

ix x t t d tφ= ⋅∫

( )y t

( ) ( ) ( )y x t h t d tτ τ= ⋅ −∫

( ) ( )sx t T t dtφ τ= ⋅ − +∫( ) ( ) ( )s iy T x t t d t xφ= ⋅ =∫

3.6. Optimal Filter: “Matched Filter” or “Correlator” – cont.

x(t) = s(t) + n(t) 0

sT

d t∫

1

sT

( )tφ

Ts MLDetectionCorrelation

Optimal Receiver dengan ML Detection untuk Binary PAM NRZ:

z

m̂Matched Filter z(t) z(kTs )

Filtering

x(t) = s(t) + n(t)

Decisionsampling st kT=

z(t) z(kTs ) m̂<

<1H

2H0z

Decision

<

<1H

2H0z

MLDetection

Correlator Receiverwith ML Detection

Matched Filter Receiverwith ML Detection

( ) ( )sh t T tϕ= −

Documents

Sistem Komunikasi II · 3.1. Introduction Encoder Modulator RF Modulator Decoder Demodulator & Detector RF Demodulator 100101… 100101… 10101… 1011… Filtering < Mapping < Detection